Figure 1411

The mean number of legal moves made by subjects in the subgoal and control conditions of Simon and Reed (1976).

the goal state. Thus, in a normative way, the theory tells us what an ideal thinker should do in this problem. From an empirical standpoint, such a normative theory allows us to look at how and why people's behaviour deviates from the ideal. We can also say what heuristic (or combination of heuristics) are the optimal ones to use to solve the problem successfully. The normative theory also allows us to determine when problems are structurally the same, even when they appear to be very different (i.e., problem isomorphs). Cognitive psychologists often borrow normative theories from other disciplines; for instance, in deductive reasoning, logic is used as the normative theory (see Chapter 16). One of the incredible things about Newell and Simon's work is that they created both the normative theory and the psychological research that followed.

Problem-space theory advances our understanding of a very complex, cognitive ability. Even though the research on puzzles is on a specific class of problems, which may be a long way from more everyday problems, it provides a solid foundation for other work. Research always has to start somewhere, and islands of understanding have to be built up. From these islands, our understanding can then extend to more complicated, real-world situations. Problem-space theory provides us with a specific account of the following:

• How people solve puzzles by applying very general rules (heuristics) to reduce the complexity of alternative solutions that are possible.

• The type of learning that can occur in problem solving; namely, the acquisition and development of different strategies for solving problems (see Anzai & Simon, 1979).

• How the understanding of a problem can affect subsequent problem-solving performance (cf. the monster-globe problems).

The problem-space account also has the following features:

• It is general enough to characterise different puzzle problems, showing the theoretical unity in many diverse instances of problem solving.

• It allows us to re-interpret previous research on other problems in an informative fashion (i.e., insight problems).

• It supports the standard model of memory with a limited-capacity working memory, that can retard problem-solving abilities (see Atwood & Polson, 1976, discussed later).

As we shall see in later chapters, these benefits extend outwards to other areas of problem solving and thinking.

Limitations of problem-space research on puzzles

Having stated the benefits that have followed from puzzle research, it is also important to be clear about the limitations of this research. There is a question mark hanging over the ecological validity of these puzzle problems. They are a special class of problems that have different properties to other problems; indeed, extensions to problem-space theory are needed to extend the generality of the theory to other classes of problems.

Puzzle problems have several contrasting properties with more mundane problems. First, puzzle problems are unfamiliar problems about which we have little knowledge (this is less the case for some insight problems, like the two-string problem). Many of the problems encountered in everyday life require considerable amounts of knowledge. Second, the knowledge required to solve puzzle problems is present in the statement of the problem. In everyday life all the information required to solve problems is often not present. In fact, much of the difficulty in everyday problems may hinge on finding the relevant information in memory or the environment required to solve the problem. If you have to buy a house, you need to know all about mortgages and current houses on sale, and finding this information is a significant part of solving the problem. Third, the requirements in puzzle problems are relatively unambiguous; the start state and goal state are clearly specified and what can and cannot be done in the problems is known (i.e., the legal moves). In everyday problems, the real difficulty may amount to specifying the nature of the goal state. For instance, doing a masters or doctoral thesis is essentially a matter of specifying where you want to end up.

In short, problem-space theory on puzzles concentrates on well defined as opposed to ill defined problems (Reitman, 1965; Simon, 1973, 1978). In well defined problems the operators, initial state, and goal state are well specified and subjects tend to have little specific knowledge about the problem. These problems tend to be solved by so-called general-purpose or domain-independent heuristics. That is, heuristics that can be applied to a wide range of situations and domains. They are rules that do not involve specific knowledge of the domain. In artificial intelligence, heuristics of this type are often called universal, weak methods. They are "universal" because they can be applied in many domains and they are "weak" because they are often not very efficient. For instance, solving one of these puzzles takes time using means-ends analysis. However, if one had rules that were specific to the problem-solving domain, the solution could be found more rapidly.

In contrast, ill defined problems can be under-specified in many ways and require the use of substantial amounts of domain-specific knowledge. The initial state of an ill defined problem may be uncertain; what is and is not part of the initial state may be unclear from the situation. If someone locks their keys inside their car, it is clear that the car and the keys locked in it are part of the initial state, but coat-hangers, brooms, the police, and owners of cars of a same make are also potentially part of the initial state. Second, the operators and operator restrictions may have to be discovered and/or created. You may have to undo some of the implicit constraints in the problem. You may have to dredge your memory for suitable operators (e.g., using a coat-hanger in a certain way, forcing a back window, finding a route into the car through the boot). Finally, the goal state may need definition. On the face of it, getting into the car is a reasonable goal state, but smashing a window to do this does not seem like a good solution; so you may want to define the goal further, to stipulate that you should get into the car without doing much damage. But what constitutes

"much damage"? Ill definition and knowledge go hand in hand because usually ill defined problems are defined through the application of knowledge. Later, we will see how people use domain-specific knowledge when they are experts in an area, based on an extension of problem-space theory. In conclusion, problemspace theory provides an adequate treatment of well defined problems, but has to be extended in order to deal with more ill defined problems.

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